Math 1090 Final Exam Fall 2012

Transcription

1 Math 1090 Final Exam Fall 2012 Name Instructor: Student ID Number: Instructions: Show all work, as partial credit will be given where appropriate. If no work is shown, there may be no credit given. All final answers should be written in the space provided on the exam and in simplified form. When needed give your answer as an exact amount, i.e., a fraction or symbolic expression like e 2, except for dollar amounts which should be rounded to the nearest cent. You may use a scientific calculator, but no cell phones, graphing or programmable calculators are allowed!! You may NOT ask questions during the exam. If you think there is an error on the exam, just write a note on that problem indicating your concern. DO NOT WRITE IN THIS TABLE!!! (It is for grading purposes.) Raw Total (out of 200 Points): Grade: Total (percentage): 1

2 1. (10 Points) Market surveys have determined that the demand for a product is p = 1 q+98 and the expected supply is p = 3q 140. If sales (supplies) 2 are to meet demand, what price should be asked for the product and how many items must be sold? price: number of items sold: 2

3 2. Find the maximum and minimum of the objective function F = 4x 3y subject to the following constraints. (You must graph the constraints and find all intersection points.) x + y 3 2x y 3 x + 2y 6 (a) (10 Points) Graph: y x (b) (5 Points) intersection points: (c) (5 Points) maximum value: minimum value: 3

4 3. Solve these equations. Give exact values for answers. (a) (10 Points) 4(2x+3) 3 2x = 2 x x = (b) (10 Points) e 2x 5e x + 2 = 2 x = 4

5 (c) (10 Points) log 2 (16) = x x = (d) (10 Points) log 2 (x + 2) = 2 + log 2 (x) + log 2 (3) (Give an exact answer, not a calculator approximation.) x = 5

6 Let A = 2 0, B = 1 5 ( ) ( ) , and C = (a) (5 Points) Determine the size of each matrix: size of A: size of B: size of C: (b) (5 Points) Is AC defined? If so, calculate it. AC = 6

7 (c) (5 Points) Is BA defined? If so, calculate it. BA = (d) (5 Points) Find A T + 3C if possible. If not possible, explain why. A T + 3C = 7

8 5. For 5x + 6y = 4 2x + 3y = 1 (a) (8 Points) Rewrite this system of linear equations Matrix form (i.e A ( x y ) = B, where A is a matrix and B is a vector). A = B = (b) (6 Points) Find A 1. A 1 = 8

9 (c) (6 Points) Solve for x and y. (x, y) = 9

10 6. Given f(x) = 3x+1 3x 2 +10x 8, g(x) = x + 1, h(x) = 2 3 x + 1, (a) (4 Points) Find the domain of f(x). domain of f(x) = (b) (4 Points) Find the domain of g(x). domain of g(x) = (c) (4 Points) Find f( 1 3 ). f( 1 3 ) = (d) (4 Points) Find (f g)( 4 3 ). (f g)( 4 3 ) = (e) (4 Points) Find h 1 (x), if possible. h 1 (x) = 10

11 7. The sock smurfs have started a business. They sneak into people s laundry rooms and steal their left socks. Then they repair the socks and sell them to the one-footed dufflepuds. The smurf business cost function is C(x) = x 2 200x and the revenue function is R(x) = 20x, where x is the number of left socks sold. (a) (5 Points) What is the smurfs profit function? P (x) = (b) (5 Points) How many left socks must the smurfs sell to break even? number of socks: 11

12 (c) (5 Points) How many left socks must the smurfs sell to maximize profit? number of socks: (d) (5 Points) What is the maximum profit? max profit: 12

13 8. For y = log 2 (x 2) + 3 (a) (6 Points) Graph the function. [Optional: You may graph the reference function on the graph below - Label your solution clearly] y x (b) (2 Points) State the domain of the function. domain: (c) (2 Points) Where is the vertical asymptote? vertical asymptote: 13

14 9. (10 Points) Prosper.com advertises that investors earn an average of 5.04% per year (compunded quarterly) by investing in loans through their website. John invests a lump sum of $18, 000 that he just received from an inheritance. How much money will John s account be worth when he retires in 25 years? account value: 10. You find a bargain house which sells for $257, 000. (a) (10 Points) What are your monthly payments for a 25-year loan at 4.25% interest compounded monthly? monthly payment: (b) (5 Points) What is the total finance charge (interest) for this 30-year loan? total finance charge: 14

15 11. (a) (10 Points) Tony and Maria have a son. They decide to create an account to save money for his college education. They want to have $200, 00 saved after 17 years. The account pays 6% interest every year, compounded twice yearly and they will make deposits at the end of each quarter. How large must each deposit be to reach their goal? quarterly deposit: (b) (10 Points) Imagine Tony and Maria have reached their goal (from part (a)). How much money can the son receive from the account twice yearly if he plans to use all the money in 4 years? quarterly withdrawal: 15